feat: Add real lorentz tensors

HepLean/SpaceTime/LorentzTensor/Real/Basic.lean

@@ -0,0 +1,88 @@
+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.SpaceTime.LorentzVector.Contraction
+import HepLean.SpaceTime.LorentzTensor.Basic
+/-!
+
+# Real Lorentz tensors
+
+-/
+noncomputable section
+
+open TensorProduct
+open minkowskiMatrix
+namespace realTensor
+
+variable {d : ℕ}
+/-!
+
+## Definitions and lemmas needed to define a `LorentzTensorStructure`
+
+-/
+inductive ColorType
+  | up
+  | down
+
+end realTensor
+
+open realTensor
+
+/-! TODO: Set up the notation `𝓛𝓣ℝ` or similar. -/
+/-- The `LorentzTensorStructure` associated with real Lorentz tensors. -/
+def realLorentzTensor (d : ℕ) : TensorStructure ℝ where
+    Color := ColorType
+    ColorModule μ :=
+      match μ with
+      | .up => LorentzVector d
+      | .down => CovariantLorentzVector d
+    τ μ :=
+      match μ with
+      | .up => .down
+      | .down => .up
+    τ_involutive μ :=
+      match μ with
+      | .up => rfl
+      | .down => rfl
+    colorModule_addCommMonoid μ :=
+      match μ with
+      | .up => instAddCommMonoidLorentzVector d
+      | .down => instAddCommMonoidCovariantLorentzVector d
+    colorModule_module μ :=
+      match μ with
+      | .up => instModuleRealLorentzVector d
+      | .down => instModuleRealCovariantLorentzVector d
+    contrDual μ :=
+      match μ with
+      | .up => LorentzVector.contrUpDown
+      | .down => LorentzVector.contrDownUp
+    contrDual_symm μ :=
+      match μ with
+      | .up => by
+        intro x y
+        simp only [LorentzVector.contrDownUp, Equiv.cast_refl, Equiv.refl_apply, LinearMap.coe_comp,
+          LinearEquiv.coe_coe, Function.comp_apply, comm_tmul]
+      | .down => by
+        intro x y
+        simp only [LorentzVector.contrDownUp, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
+          comm_tmul, Equiv.cast_refl, Equiv.refl_apply]
+    unit μ :=
+      match μ with
+      | .up => LorentzVector.unitUp
+      | .down => LorentzVector.unitDown
+    unit_lid μ :=
+      match μ with
+      | .up => LorentzVector.unitUp_lid
+      | .down => LorentzVector.unitDown_lid
+    metric μ :=
+      match μ with
+      | realTensor.ColorType.up => asProdLorentzVector
+      | realTensor.ColorType.down => asProdCovariantLorentzVector
+    metric_dual μ :=
+      match μ with
+      | realTensor.ColorType.up => asProdLorentzVector_contr_asCovariantProdLorentzVector
+      | realTensor.ColorType.down => asProdCovariantLorentzVector_contr_asProdLorentzVector
+
+end